**Ref:** R. W. Rollins, P. Parmananda, and P. Sherard,
Phys. Rev. E, **47**, R780-R783 (1993).

**
CONTROLLING CHAOS IN HIGHLY DISSIPATIVE SYSTEMS:**

a simple recursive algorithm

**
**
R. W. Rollins, P. Parmananda, and P. Sherard

*Condensed Matter and Surface Sciences Program,*

Department of Physics and Astronomy,

Ohio University, Athens, Ohio 45701-2979

### Abstract

We present a new recursive proportional-feedback (RPF) algorithm for
controlling deterministic chaos. The algorithm is an adaptation of
the method of Dressler and Nitsche
[Phys. Rev. Lett. **68** 1 (1992)]
to highly dissipative systems with dynamics that show a
nearly one-dimensional
return map of a single variable, $X$, measured at
each Poincar\'{e} cycle.
The result extends the usefulness of simple proportional-feedback
control algorithms.
The change in control parameter prescribed for the
$n$--th Poincar\'{e} cycle by the RPF algorithm is
given by $\delta p_n = K (X_n-X_F) + R \delta p_{n-1}$, where $X_F$
is the fixed point of the target orbit, and
$K$ and $R$ are proportionality constants.
The recursive term is shown to arise
fundamentally because, in general, the Poincar\'{e}
section of the attractor near $X_F$ will change position in
phase space as small changes are made in the control parameter.
We show how to obtain $K$ and $R$ from simple
measurements of the return map without any prior knowledge of
the system dynamics and report the successful application of the RPF
algorithm to model systems from chemistry and biology
where the recursive term is necessary to achieve control.

PACS Nos.: 05.45.+b, 06.70.Td, 87.10.+e

**Ref:** R. W. Rollins, P. Parmananda, and P. Sherard,
Phys. Rev. E, **47**, R780-R783 (1993).

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